This is a multi-part message in MIME format. ---------------------- multipart/alternative attachment Hello list Like all of you, I imagine, I was first made aware of these beats when = listening to a piano (in my case pipe organ) being tuned. My curiosity = led me to analyse the reason and I turned to simple mathematics for the = answer. Enter my "small equation" where the fundamental (1st.Harm.) is = multiplied by the 12th. root of 2 to derive the next fundamental in the = ascending equal tempered chromatic scale. (Simply devide if you want to = go down chromatically). In every case the resultant becomes a = Fundamental Note - or 1st.Harmonic. I know this is all "old hat" but = there's probably some meat in it to be chewed over just the same. Now = taking the Fundamental and multiplying by "Two" one gets the second = harmonic. Multiplying by "Three" gets the third harmonic ad. inf. The = really interesting point now comes... Take your M10 based on A# (which = computes at 466.16376.....)The 5th. harmonic of that A# =3D = 2796.9826...... the M10 is F(which computes at 1396.9129.......) The = 2nd. harmonic of that F =3D2793.8259...... = There we have a difference of 3.1567..... Hz which is the beat you hear. = The same applies, of course, to any and all intervals in the equal = tempered scale, be they M3,M4,M5 - whatever. It is interesting to apply that "small formula" to log all the notes of = the equal tempered scale on a mathematical basis and thereby to extract = the "Beat-rate" information whole-sale. To me this information is "an = invisible tool" - and I use it - as do you. But it's nice to see an = invisible friend sometimes!=20 Of course! There are many pitfalls! Many pianos do not produce the exact = mathematical frequencies according to my table. Those times when the = first covered (wrapped) Bass string are played with their M10 - and the = beat rate does not conform to the mathematical progression of the tables = as produced in the "metals" section of the piano. Well..., That's life = :-) I find the "scientific calculator on the PC best as it produces so = many decimal places! In using such a calculator you can go right up the = scale and land up, an 8ve later with exactly twice the number you = started out with. Now that's Magic!=20 Regards Michael G (UK) ---------------------- multipart/alternative attachment An HTML attachment was scrubbed... URL: https://www.moypiano.com/ptg/pianotech.php/attachments/8e/e9/68/15/attachment.htm ---------------------- multipart/alternative attachment--
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